On spectral properties of positive operators

[摘要] This thesis deals with the spectral behavior of positive operators and related oneson Banach lattices. We first study the spectral properties of those positive operatorsthat satisfy the so-called condition (c). A bounded linear operator T on a Banachspace is said to satisfy the condition (c) if it is invertible and if the number 0 is inthe unbounded connected component of its resolvent set p(T). By using techniquesin complex analysis and in operator theory, we prove that if T is a positive operatorsatisfying the condition (c) on a Banach lattice E then there exists a positive number aand a positive integer k such that T^k ≥ a•I, where I is the identity operator on E. Asconsequences of this result, we deduce some theorems concerning the behavior of theperipheral spectrum of positive operators satisfying the condition (c). In particular,we prove that if T is a positive operator with its spectrum contained in the unit circleΓ then either σ(T) = Γ or σ(T) is finite and cyclic and consists of k-th roots of unityfor some k. We also prove that under certain additional conditions a positive operatorwith its spectrum contained in the unit circle will become an isometry. Another mainresult of this thesis is the decomposition theorem for disjointness preserving operators.We prove that under some natural conditions if T is a disjointness preserving operatoron an order complete Banach lattice E such that its adjoint T' is also a disjointnesspreserving operator then there exists a family of T-reducing bands {E_n : ≥ 1} U{E_∞} of E such that T|E_n has strict period n and that T|E_∞ is aperiodic. We alsoprove that any disjointness preserving operator with its spectrum contained in asector of angle less than π can be decomposed into a sum of a central operator and aquasi-nilpotent operator. Among other things we give some conditions under whichan operator T lies in the center of the Banach lattice. Also discussed in this thesisare certain conditions under which a positive operator T with σ(T) = {1} is greaterthan or equal to the identity operator I.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics